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Why shall we?
As usual, the best storage solution is first to avoid storage. It applies also to network communication, database access, etc. Often used as a first slide of a presentation, but you still need a follow up.
In the case of energy storage, demand-shifting is entailed:

- Supply side economic incentives (heavily variable pricing)

- Accessible, standardised real-time and forecast information about rates

- Client side automated usage control (with defaults)

This could be a good application for IoT. For instance, the only use-case I'd seriously consider for an internet enabled fridge, would be if it could "store" energy in the form of deep-chilling a thermal insulator to be charged at lower rates, when it's windy say.

One way of thinking about energy storage is traditional fossil fuels, with a whole speculator's oil market attached to it.

There is another way of thinking. When one barely have a shortage of something, like spring water, which flows from mountains (and fluctuates heavily depending on season, monsoon, etc) one probable doesn't store much and let it flow.

For wind and solar, it seems, this is much better paradigm. Send it to the grid when its available, get it from the grid when it is still and cloudy.

Water pumps.
You need very particular geography for pumped hydro, plus in hot dry climates your stored energy literally evaporates.
A more general solution: just use weights https://www.gravitricity.com
Be interesting to see how you could build this on a smaller scale for home use, using something very heavy but cheap like lead.
Lead is actually not that cheap. Lead ingots are $6/pound (at McMaster), while steel is $.50/pound (from Alibaba, but that's a minimum purchase of 1 ton). Scrap lead is around $.40/pound, versus steel at $.02/pound. The cheapest thing would probably be rock/concrete, or water.
By cheap I meant not having to dig a tunnel, by using lead, which I think is the cheapest dense metal?, you avoid having to have a very long tether, say compared to water/concrete.
A clip and pulley system which could hook onto a barbell or kettle bell would be great.

Hrmm, perhaps this isn't so feasible. Performing a clean & jerk of 95 pounds to a height of 7 feet would yield about 900 watt seconds of potential energy. So even if you had a very efficient laptop (15 watts) and perfect energy conversion, you'd have to perform a clean & jerk every 60 seconds to keep computing. Even if you were in good shape, you probably couldn't keep that up for more than a hour.

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

"to get the amount of energy stored in a single AA battery, we would have to lift 100 kg (220 lb) 10 m (33 ft) to match it. To match the energy contained in a gallon of gasoline, we would have to lift 13 tons of water (3500 gallons) one kilometer high (3,280 feet). It is clear that the energy density of gravitational storage is severely disadvantaged." http://physics.ucsd.edu/do-the-math/2011/11/pump-up-the-stor...
Then how about sewage pumps For a combined energy storage and water distillation facility?
Then how about sewage pumps For a combined energy storage and water distillation facility?
Molten salt batteries are actually really good for this kind of usage. High power output and cycle life are key requirements, constant usage means energy usage for thermal maintenance is less of a factor, and the fact that it's stationary means energy density and impact resistance are less important.
I haven't researched the overhead costs, but there are alternative kinetic "batteries" that have been proposed, some of which are already being used:

- Use excess energy generation to drive a gravel-filled train up a few miles of 2% grade, back down the track to drive turbine(s) for energy to be tapped

- Run water up an incline into a reservoir. This can simultaneously serve irrigation purposes. Water level is indicative of available energy to be sent through turbines

- Multi-ton flywheels. I've seen that some fusion experiments use this for massive, quick energy bursts. Magnetism prevents excessive wear/tear on bearings

composite thin cylindrical flywheels have the best energy storage characteristics

"greater than 400 Wh/kg can be achieved by certain composite materials"

https://en.m.wikipedia.org/wiki/Flywheel_energy_storage

Flywheels sound exciting but their energy density is about the same as Li Ion batteries (even when carbon composites are used). There are very few large scale deployments of flywheel energy storage systems in the world despite lots of active interest in them. Safety is also a huge concern since if there is a mechanical failure, a huge amount of kinetic energy will have to be safely disposed or you are going to have shrapnels flying everywhere.
You can recharge the flywheel over 10M times. Containment is designed in.
In certain failure scenarios I think an energetic flywheel is predictable: Most of the momentum is within the plane of rotation, which is where you can place whatever kind of shielding or catching-material.
Flywheel systems last considerably longer than batteries, and are generally installed dug into the ground where failures cause no safety issues. Even when not installed underground the vessels fully contain any failure in any case.
Maybe store it in electric car batteries? The ppl already buying Tesla Model 3 at scale.

Why utilities should pay for storage, when they could just offer cheaper electricity when there is excess of it. E.g. charge your car when we told you to and save %30 on your bill!

Yes, this is one of the classic suggestions for overnight or at-work charging: cars which choose to charge depending on the owner's goals and energy prices.
Not exactly storing energy, but a functional equivalent is load shifting. This means shifting demand from when the electricity supply is low to when it is high. An awful lot can be done with this.

This can be implemented by adjusting electricity rates in real time, and having loads that can be shifted monitoring the rates and turn on when they are low.

I live in Shizuoka, Japan where a lot of cars and motorcycles are built. When the nuclear reactors were shut down one of the ways they managed the reduction in power usage was to have the factories cooperate on their work schedules. So Suzuki would open from 9-5 and then Yamaha would open, etc. It only lasted for a few months, but I've heard it made a huge difference in helping the grid meet demands.
That makes a lot of sense. Our current system of a flat rate for power regardless of demand is ridiculously wasteful.

For example, consider an electric hot water heater. The water will stay hot for a day in a well insulated one. Having it set to run only when rates are lower would to a lot to match supply with demand, reduce pollution, costs, etc.

A number of Chicago skyscrapers are cooled with ice made during off-peak hours. The OP's article mentions ice as some new technique. It's been done in production for a long time now.

http://enwavechicago.com/

"Nice"? The whole world will be running on solar and wind electricity in 20 years. "Nice" is a bit of an understatement.
Hardly. There are fossil and nuclear plants being built today with design lifetimes of 20 years. Adoption is certainly shooting upwards in some countries, but it's going to be a while before renewables go over 50% worldwide.
LightSail energy [0] has been working on this problem for the past five years. They store the energy in (surprise!) highly efficient, long-term storage air compressors.

[0] crunchbase.com/organization/lightsail-energy -- $42M funded

By and large, don't. Let the market for electricity react to fluctuations in output via variable pricing.

This is already being done to a certain extent in Germany (aluminum smelters will scale production up or down based upon the supply of electricity).

Over time as fluctuations in electricity supply from renewables become more common, the demand side will react. Those periods when the wholesale price of electricity drop to zero won't recur forever as somebody enterprising will make use of it. It won't happen overnight but it will happen.

It's ironic that the same industry (utilities) that told us the market was a solution to all of our energy problems when it wasn't (see Paul Krugman's defence of Enron's bullshit in the early 00s) is now conveniently forgetting that canard when it's actually true for once because renewables threaten its bottom line.